Formation of Photoelectron Holograms of Crystals Excited by Synchrotron Radiation V
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Crystallography Reports, Vol. 47, No. 2, 2002, pp. 165–173. Translated from Kristallografiya, Vol. 47, No. 2, 2002, pp. 199–207. Original Russian Text Copyright © 2002 by Nosik. DIFFRACTION AND SCATTERING OF X-RAY AND SYNCHROTRON RADIATION Formation of Photoelectron Holograms of Crystals Excited by Synchrotron Radiation V. L. Nosik Center of Technological Innovations Crystal Growth, Russian Academy of Sciences, LeninskiÏ pr. 59, Moscow, 117333 Russia Received June 15, 2001 Abstract—The theoretical aspects of formation of photoelectron beams excited by the ultraviolet synchrotron radiation incident onto a crystal are considered. It is shown that the change in the energy of the monochroma- tized synchrotron radiation incident onto a hematite crystal results in the change of the number of reflections participating in the hologram formation. © 2002 MAIK “Nauka/Interperiodica”. INTRODUCTION SPECIMEN The development of nanotechnology resulted in the The specimen was an orthorhombic hematite creation of various structural methods of visualizing α -Fe2O3 crystal with the unit-cell parameter a = 5.42 Å atomic structures which are based on the use of short- and α = 55.17° cut out in such a way that the hexagonal wavelength radiation (neutrons, electrons, X-ray and c-axis was normal to the entrance surface (Fig. 1). This synchrotron radiation, etc.). The term atomic hologra- crystal was selected because of the characteristic mag- phy coined by Gabor in 1948 [1] describes the method netic structure of the outer electron shells and its simple of the three-dimensional reconstruction of atomic structures due to positive interference of the strong ref- atomic structure. erence and weak object waves [2]. Later, the terms It is well known that the outer electrons of an iron atomic holography or the Fourier-transformation with- 6 2 atom (3d 4s ) obey the inequality E4s < E3d, which out a lens became widely used because of the well- explains the incomplete filling of the d-shell. It is inter- known optical analogy. esting to analyze the distributions of both collective However, the principle of image formation was 3d-electrons (magnetic structure) and 4s-electrons known long ago, e.g., as Kikuchi lines (in X-ray and (positions of individual atoms) in these crystals. Below, electron diffraction), X-ray standing waves in multi- we consider only the photoeffect of 4s-electrons. beam diffraction, Kossel lines, etc. [3, 4]. Lately, most of the published articles on this subject considered electron holography [5–7]. It was shown that, for high-energy electrons, the Cowley–Moody cal- culations [8] yield images whose main features are the same as those of the experimental images [4]. However, p strong scattering of secondary electrons considerably z, c distorts the signal, which, in turn, hinders image pro- cessing. x At the same time, some schemes for studying photo- a1 and Auger electrons, elastically scattered X-ray radia- tion, fluorescence, etc. were created [9]. a y a Below, we consider the theoretical aspects of one of k 2 the new methods—positional-sensitive photoelectron spectroscopy—where excitation is caused by beams of soft synchrotron radiation. In fact, this method is a modern modification of photoelectron spectroscopy with angular resolution [10], which has been widely used for a long time to determine the band structure of Fig. 1. Geometry of scattering from an oriented hematite electrons. crystal. 1063-7745/02/4702-0165 $22.00 © 2002 MAIK “Nauka/Interperiodica” 166 NOSIK THEORY The coefficient B(p, t) determines the dependence of Within the framework of perturbation theory [11], the scattered radiation on time absorption of a photon with the potential vector A, the 2 ω expi()បω – E – p /2m – 1 wave vector Q, and the frequency (in the dipole Bpt(), = -------------------------------------------------------------------4s -. (8) approximation exp(iQa) ≈ 1, V = 1), បω 2 – E4s – p /2m 2π A = ------ eexp()–iωt (1) When SR action on the atom is prolonged, the coef- ω ficient B(p, t) is simplified to the form results in electron transition (with a certain probability) π 2 from the initial state ψ localized at the atom R to one (), 2 2 tδp 2 4s i lim Bpt = -------- ------- – k of the delocalized states, t → ∞ ប 2m (9) π ψ ()ψrR– , t = ()ψrR– + 0, 2 tm()δ()δ() i i 4s i i = --------ប ---- pz – kz + pz – kz (2) kz ψ 0 3 () (), i = ∫d piexp pr Ui B p t , and has the nonzero value only if the law of energy con- where the state of the photoelectron is described by the servation is obeyed. We used the following notation: integral over the plane waves exp(ipr), whereas the pulse p is measured in units ប. 2 បω 2 2 k ==+2E4s, kz mk –,pt (10) It should be emphasized that the calculation of the matrix element Ui of the transition where pt and pz are the components of the pulse vector lying in the plane of the crystal surface and normal to it, π 2 h ψ ()()∇ ()3 respectively. Ui = –i ------ω ----∫ 4s rR– i e exp –ipr d r (3) m Since the photoeffect is an inelastic process, no results in the appearance of the phase factor interference between the waves of the primary photo- electrons emitted by various atoms is possible, which () () Ui = epDpexp ipRi , (4) can readily be seen if one takes into account the arbi- where e is the polarization vector of the incident syn- trary phase factors before the wave functions of the core chrotron radiation (SR), and p, the wave vector of the electrons. The holographic effects can be explained if photoelectron. one takes into account the interference between the pri- mary photoelectron radiation and the elastically scat- The coefficient D(p) is independent of the direction tered secondary photoelectron waves. The electron flux of the vector p because of the spherical symmetry of the ψ through the unit area J recorded by a two-dimensional wave function 4s. Thus, in the hydrogen-like approxi- detector (the surface normal n) is determined as mation we have ()φ ប ψ exp i 3/2 J = ------()ψ*()ψψn∇ – ()ψn∇ * . (11) 4s = ------------------- Z im 44π (5) Using expression (2) and ignoring photoelectron ×ξ()ξ ξ2 2 1ξ3 3 exp – r 13–2r + r – --- r , scattering, we arrive at the following expression for 3 intensity, which has no term containing information on ξ the structure, where = Z/4aB, Z is the charge of the iron nucleus, the ប2 2 distances are measured in atomic units aB = /me = ប 0.529 Å, and φ is an arbitrary phase of the wave func- ∆ ∆ 3 () 2 (), 2τ () J0 S = S----∑∫d p np Ui Bpt i p , (12) tion. Differentiating the well-known expression m i 3 exp()– λr + ipr 4π τ ∫d r--------------------------------------- = ----------------- (6) where i(p) takes into account inelastic scattering and r p2 + λ2 absorption of photoelectrons prior to their detection, ∆ with respect to λ, we arrive at the cumbersome but rig- and S is the area of the detector pixel. orous expression for the function D(p), Now, take into account the scattering of the primary photoelectron wave propagating from the ith and scat- 1 Dp()∼ exp()iφ -----------------()ξ…+ . (7) tered by the jth atoms with the scattering field ϕ(r) p2 + ξ2 using perturbation theory (the kinematical or the Born approximation). The secondary wave generated by this Taking into account the high degree of monochro- atom is the convolution matization of the incident synchrotron radiation, we assume the pulse p and the coefficient D(p) to be con- ψ s () 3 ()ϕ()ψ0(), stant. ij R0 = –∫d rG R r j + r i r j + r t , (13) CRYSTALLOGRAPHY REPORTS Vol. 47 No. 2 2002 FORMATION OF PHOTOELECTRON HOLOGRAMS 167 where R = R0 Ð rj Ð r. Expanding the wave function of we can represent the wave function of photoelectrons the primary photoelectrons, (2), and the Green’s func- scattered by all the atoms in the form tion into the Fourier series s 1 exp()ikR 3 exp()iqr ψ ()R = exp()–iH R GR()==–------ ----------------------- d q---------------------- , i 0 ∑ jk 0 π ∫ 2 2 (20) 4 R q – k (14) jk 3 z 2 2 × (), ()(), k = p /2m, ∫d pUi Bptexp iptR0 F jk p R0 , we arrive at the following expression for the secondary where wave: 2 z 4π τ()p – q 3 () z z F jk p; R0 = -------- ∫dqz------------------------------------------------ ψ s () 3 (), d q 2 2 ()2 2 ij R0 = ∫d pUi Bpt∫---------------- a qz + pt – H jk – k q2 – k2 (15) × ()()z (), × ()()()π () exp iqz – pz R0 f H jk pz – qz . exp iqR0 exp i pq– r j 4 f pq– , The function τ p q describes the effective obtained based on the definition of the structure fac- ( z Ð z) tor [12] decrease of the contributions from deeply located atomic planes f ()pq– = ∫d3rϕ()r exp()i()pq– r . (16) Z τ()p – q = ∑ exp – ip()– q Z – -----s , (21) Assuming that the atom is spherically symmetric, z z z z s L we obtain the atomic factor in the form s where L is the free path, which for electrons with the 2me2 Zf– ()s f ()s = ------------ ----------------------p , (17) energy ranging within 10–1000 eV is almost constant ប2 s2 and equals several interatomic distances, and L ≈ 10 Å. Thus, only several upper atomic planes actively partic- where the term proportional to the number of protons, ipate in the signal formation. Z, corresponds to scattering from the nucleus and the With due regard for elastic scattering, the flux of term fp(s), to scattering by the electron shells of the electrons from the ith atom is the sum of the primary atoms.